Light: Reflection and Refraction

Light travels in straight lines. When it falls on a polished surface such as a mirror it bounces back, and this is reflection. The chapter opens by restating the two laws of reflection you met in Class 8, because everything about spherical mirrors is built on them.

Laws of reflection: (i) the angle of incidence is equal to the angle of reflection, ∠i = ∠r; (ii) the incident ray, the normal at the point of incidence and the reflected ray all lie in the same plane. Both angles are measured from the normal, never from the mirror surface. These laws hold for every reflecting surface, flat or curved, because a curved mirror is treated as a very large number of tiny plane mirrors.

A plane mirror forms an image that is virtual and erect, of the same size as the object, at the same distance behind the mirror as the object is in front of it, and laterally inverted (left appears right). These four characteristics are asked as a 2-mark question on their own. The magnification of a plane mirror is +1: the plus sign says erect and virtual, the 1 says same size.

Two words to fix now. A real image is formed when reflected (or refracted) rays actually meet; it can be caught on a screen and is always inverted. A virtual image is formed where rays only appear to meet after being produced backwards; it cannot be caught on a screen and is always erect. Every later question about the nature of an image is answered with these two words plus size.

For a ray falling along the normal, the angle of incidence is 0°, so the angle of reflection is 0° and the ray retraces its path. This one fact explains why a ray through the centre of curvature of a spherical mirror reflects back along the same line, which is the third ray you will use in every mirror diagram.

A spherical mirror is a mirror whose reflecting surface is a part of a hollow sphere. If the reflecting surface curves inwards (towards the centre of the sphere) it is a concave mirror; if it curves outwards it is a convex mirror. A concave mirror converges parallel rays, a convex mirror diverges them, so you will also hear the names converging mirror and diverging mirror.

Six terms describe any spherical mirror and the examiner expects the exact NCERT wording.

• Pole (P): the centre of the reflecting surface. It lies on the surface of the mirror.
• Centre of curvature (C): the centre of the sphere of which the mirror is a part. It lies in front of a concave mirror and behind a convex mirror, so it is not a part of the mirror.
• Radius of curvature (R): the radius of that sphere, that is, the distance PC.
• Principal axis: the straight line passing through the pole and the centre of curvature. It is normal to the mirror at the pole.
• Principal focus (F): for a concave mirror, the point on the principal axis where rays parallel to the principal axis actually meet after reflection. For a convex mirror, the point on the principal axis from which the reflected rays appear to diverge. So F is real for a concave mirror and virtual for a convex mirror.
• Focal length (f): the distance between the pole and the principal focus, PF.
• Aperture: the diameter of the reflecting surface. NCERT only considers mirrors whose aperture is much smaller than the radius of curvature.

For such small-aperture mirrors the radius of curvature is twice the focal length: R = 2f. A one-line question like the radius of curvature of a spherical mirror is 20 cm, what is its focal length, is answered f = R/2 = 10 cm.

The NCERT activity with a concave mirror and sunlight is worth knowing. Hold the mirror facing the sun and move a sheet of paper until you get the sharpest, brightest spot; that spot is the image of the sun at the principal focus, and the distance from the mirror to the paper is roughly the focal length. The paper eventually chars, which is why the same idea builds a solar furnace.

Before drawing anything, learn the table of images formed by a concave mirror for the six standard object positions. It is the single most reused piece of this chapter, and every numerical answer should agree with it.

• Object at infinity: image at F, highly diminished, point-sized, real and inverted.
• Object beyond C: image between F and C, diminished, real and inverted.
• Object at C: image at C, same size, real and inverted.
• Object between C and F: image beyond C, enlarged, real and inverted.
• Object at F: image at infinity, highly enlarged, real and inverted.
• Object between P and F: image behind the mirror, enlarged, virtual and erect.

The pattern is easy to remember: as the object moves from infinity towards F, the image moves from F outwards to infinity, growing all the way, and stays real and inverted. Only when the object crosses inside F does the image flip to the other side, becoming virtual, erect and magnified. That last case is the one used by dentists, in shaving mirrors and in make-up mirrors.

A convex mirror has only two cases to learn. Object at infinity: image at F behind the mirror, highly diminished, point-sized, virtual and erect. Object anywhere between infinity and the pole: image between P and F behind the mirror, diminished, virtual and erect. Whatever you do, a convex mirror never gives a real image and never gives an enlarged one. This is the basis of the frequently asked question, which mirror always forms a virtual, erect and diminished image; the answer is a convex mirror.

The NCERT activity behind the table uses a candle flame and a concave mirror on a table, with a screen moved until a sharp image appears. Place the candle at different positions beyond C, at C, between C and F, at F and inside F, and note where and how large the image is. You may be asked to describe this activity or to complete an observation table from it.

A ray diagram uses two of the following four rays; any two are enough because all rays from a point on the object meet (or appear to meet) at the same image point.

1. A ray parallel to the principal axis passes through the principal focus after reflection from a concave mirror (or appears to diverge from F for a convex mirror).
2. A ray passing through the principal focus of a concave mirror (or directed towards F of a convex mirror) becomes parallel to the principal axis after reflection.
3. A ray passing through the centre of curvature of a concave mirror (or directed towards C of a convex mirror) is reflected back along the same path, because it hits the mirror along the normal.
4. A ray incident obliquely at the pole is reflected obliquely, making equal angles with the principal axis (which is the normal at P).

How to draw for full marks: draw the principal axis, mark P, F and C on it with C at twice the distance of F, draw the mirror as a curve with small hatch marks on the non-reflecting side, draw the object as an upright arrow standing on the axis, take rays 1 and 3 (or 1 and 2) from the tip of the arrow, put arrowheads on every ray, and draw the image arrow where the reflected rays meet. For the object between P and F, the reflected rays diverge, so extend them backwards as dotted lines behind the mirror and draw the image dotted too, to show it is virtual. Label the image and state its nature below the diagram.

Uses of a concave mirror (a 2- or 3-mark favourite, always with a reason): in torches, searchlights and vehicle headlights, where the bulb is placed at F so the reflected light comes out as a powerful parallel beam; as shaving mirrors and make-up mirrors, held closer than F so the face appears erect and enlarged; by dentists, who hold the mirror within the focal length to see a large erect image of a tooth; and in solar furnaces, where a large concave reflector focuses sunlight on the furnace placed at its focus to produce high temperature.

Only two diagrams exist for a convex mirror, and both give a virtual, erect, diminished image behind the mirror. Draw the axis, P on the mirror, then F and C behind the mirror (C at twice the distance of F). From the tip of the object arrow take a ray parallel to the axis; after reflection it goes away from the mirror as though it came from F, so draw the reflected ray and extend it backwards as a dotted line through F. Take a second ray aimed at C; it hits the mirror along the normal and reflects straight back, so its backward extension passes through C. The two dotted extensions meet between P and F, and that is the image, drawn dotted. If the object is at infinity, the parallel rays reflect so that their backward extensions all pass through F, and the image is a point at F.

Questions of the form draw the path of the reflected ray for an incident ray parallel to the principal axis, or for a ray directed towards the principal focus, or towards the centre of curvature, of a convex mirror, are single-ray diagrams and have appeared repeatedly (2014, 2019, 2023). For the ray aimed at F the reflected ray comes out parallel to the axis; for the ray aimed at C it retraces its path.

Uses of convex mirrors. They are used as rear-view (wing) mirrors in vehicles, and this is the most asked reason question in the chapter. Two points earn the marks: a convex mirror always gives an erect image, so the driver sees traffic the right way up, and because the image is highly diminished it has a wide field of view, so the driver sees a much larger area behind than a plane mirror of the same size would show. Convex mirrors are also fixed on the sides of big shops and at sharp bends on hilly roads for the same wide-field reason, and as security mirrors in stores.

The NCERT activity is worth a line: hold a convex mirror in front of you, then move it away, and the image of your face stays erect and gets smaller while more of the room appears in it. If you look at the back of a shining steel spoon you see the same effect; the front of the spoon acts as a concave mirror and shows an inverted image when held at arm's length.

Every numerical in this chapter depends on the New Cartesian Sign Convention. Learn its five rules as a list; they are asked directly for 2 marks and they decide the sign of every number you substitute.

1. The object is always placed to the left of the mirror, so light from the object falls on the mirror from the left-hand side.
2. All distances parallel to the principal axis are measured from the pole of the mirror.
3. Distances measured to the right of the pole, in the direction of the incident light, are taken as positive; distances measured to the left of the pole, against the incident light, are negative.
4. Distances measured perpendicular to and above the principal axis are positive.
5. Distances measured perpendicular to and below the principal axis are negative.

What this means in practice. The object distance u is always negative, because the object is on the left. The focal length of a concave mirror is negative (F is in front, to the left), and the focal length of a convex mirror is positive (F is behind, to the right). The radius of curvature carries the same sign as the focal length. A real image formed in front of a concave mirror has negative v; a virtual image behind any mirror has positive v. Object height h is positive; an inverted image has negative height, an erect image has positive height.

This is why the MCQ a spherical mirror has a focal length of −15 cm, what is it, is answered concave mirror, and why a mirror with focal length +15 cm is convex. It is also why you never carry the minus sign into the answer sentence: when you get v = −30 cm, you write that the image is formed 30 cm in front of the mirror, on the same side as the object, and it is real. The sign tells you the side; the sentence tells the examiner you understood it.

The mirror formula links the object distance u, the image distance v and the focal length f, all measured from the pole with the sign convention: 1/v + 1/u = 1/f. It applies to both concave and convex mirrors in all situations, provided the signs are put in correctly. Class 10 does not ask you to derive it, but you must be able to use it fluently.

Magnification m is the ratio of the height of the image to the height of the object: m = h′/h. For a spherical mirror it is also given by m = −v/u. Both forms are needed; the second lets you find the size of the image from the distances. The sign of m carries information: m is negative for a real, inverted image and positive for a virtual, erect image. The magnitude tells the size: |m| greater than 1 means enlarged, equal to 1 means same size, less than 1 means diminished. So a magnification of −3 for a concave mirror means a real, inverted image three times the size of the object, and +1/3 for a convex mirror means a virtual, erect image one-third the size.

A worked example in the style of the board. An object 4 cm tall is placed 25 cm in front of a concave mirror of focal length 15 cm. Data: h = +4 cm, u = −25 cm, f = −15 cm. From 1/v = 1/f − 1/u = −1/15 + 1/25 = (−5 + 3)/75 = −2/75, so v = −37.5 cm. The image is 37.5 cm in front of the mirror, real and inverted. m = −v/u = −(−37.5)/(−25) = −1.5, so h′ = m × h = −1.5 × 4 = −6 cm: an inverted image 6 cm tall. Check against the table: the object is between C (30 cm) and F (15 cm), so the image should be beyond C, real, inverted and enlarged. It is.

A second one for a convex mirror. An object is 30 cm in front of a convex mirror of focal length 15 cm. u = −30 cm, f = +15 cm. 1/v = 1/15 + 1/30 = 3/30 = 1/10, so v = +10 cm, behind the mirror. m = −v/u = −10/(−30) = +1/3. The image is virtual, erect and one-third the size, 10 cm behind the mirror, exactly what a convex mirror always does.

Light travels at different speeds in different media. When it passes obliquely from one transparent medium into another, its direction changes at the boundary; this bending is refraction, and the cause is the change in speed. Familiar examples: the bottom of a tank of water appears raised, a pencil partly dipped in water looks bent at the surface, a lemon in a glass of water looks larger, and printed letters seen through a thick glass slab appear raised.

Optical density is the word used to compare media. A medium in which light travels slower is optically denser, and one in which it travels faster is optically rarer. This is not the same as mass density: kerosene is less dense than water by mass but is optically denser than water (its refractive index is higher). A 2024 question was built entirely on this distinction.

Direction of bending. When light goes from a rarer medium into a denser medium (air to water, air to glass) it slows down and bends towards the normal, so the angle of refraction is smaller than the angle of incidence. When it goes from a denser to a rarer medium (glass to air) it speeds up and bends away from the normal, so the angle of refraction is larger. A ray that hits the boundary along the normal (angle of incidence 0°) goes straight through without bending, although its speed still changes.

Laws of refraction, both to be written every time they are asked: (i) the incident ray, the refracted ray and the normal to the interface at the point of incidence all lie in the same plane; (ii) the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media and a given colour of light. The second law is Snell's law, sin i / sin r = constant, and the constant is the refractive index of the second medium with respect to the first. In the board paper, laws of refraction plus the definition of absolute refractive index is a standard 3-mark question (2015, 2018, 2024).

The NCERT activity fixes a glass slab on white paper, marks a line as the incident ray with two pins, and finds the emergent ray by placing two more pins on the far side so that all four appear in one line. Joining the points gives the path through the slab, and you measure three angles: the angle of incidence i at the first surface (air to glass), the angle of refraction r inside the slab, and the angle of emergence e at the second surface (glass to air).

What is found. At the first surface the ray bends towards the normal (r is less than i), because it enters the denser medium. At the second surface it bends away from the normal because it re-enters air. The two surfaces are parallel, so the angle at which the ray meets the second surface equals r, and by the reversibility of light the angle of emergence equals the original angle of incidence: ∠e = ∠i. Therefore the emergent ray is parallel to the incident ray, but it is shifted sideways. The perpendicular distance between the incident ray produced and the emergent ray is called the lateral displacement (or lateral shift).

A common error is to write that there is no refraction in a glass slab because the ray comes out parallel. Refraction happens twice; the two bendings are equal and opposite, which is why the direction is restored while the position is not. The lateral displacement increases with the thickness of the slab, with the angle of incidence, and with the refractive index of the glass. For a ray incident normally, i = 0, so r = 0 and e = 0, and the ray goes straight through with no lateral shift.

Exam forms of this section: draw the path of a ray through a glass slab and label i, r, e and the lateral displacement (2013, 2016); a ray of light incident at 45° on one face of a slab emerges at what angle from the other face (45°, since e = i); the device that produces lateral displacement without changing direction (a rectangular glass slab, 2025 MCQ); and an assertion-reason on the emergent ray being parallel.

The refractive index of medium 2 with respect to medium 1 is the ratio of the speed of light in medium 1 to the speed of light in medium 2: n₂₁ = v₁/v₂. It equals the constant in Snell's law. Similarly, the refractive index of medium 1 with respect to medium 2 is n₁₂ = v₂/v₁, so n₁₂ = 1/n₂₁. This relative refractive index is what you measure between, say, water and glass.

When medium 1 is vacuum (or air, to a very good approximation), the refractive index is called the absolute refractive index of the medium: n_m = c/v, the speed of light in vacuum divided by the speed of light in the medium. Since light is fastest in vacuum, v is always less than c and the absolute refractive index is always greater than 1 (it is exactly 1 for vacuum and 1.0003 for air). It has no unit, being a ratio of two speeds. Saying the refractive index of diamond is 2.42 means light travels 2.42 times slower in diamond than in vacuum, or that c/v = 2.42 with v = 3 × 10⁸/2.42 ≈ 1.24 × 10⁸ m/s.

The values from NCERT Table 9.3 that get used in questions: air 1.0003, ice 1.31, water 1.33, alcohol 1.36, kerosene 1.44, turpentine 1.47, crown glass 1.52, benzene 1.50, rock salt 1.54, dense flint glass 1.65, ruby 1.71, sapphire 1.77, diamond 2.42. Diamond has the highest optical density and air the lowest. Of kerosene, turpentine and water, light travels fastest in water because water has the lowest refractive index of the three.

The two standard numericals. First: the refractive index of glass is 1.5, find the speed of light in glass. v = c/n = 3 × 10⁸/1.5 = 2 × 10⁸ m/s. Second (2016 and often since): the refractive indices of glass and water with respect to air are 3/2 and 4/3; if the speed of light in glass is 2 × 10⁸ m/s, find the speed in water. First recover c = n_g × v_g = 1.5 × 2 × 10⁸ = 3 × 10⁸ m/s, then v_w = c/n_w = 3 × 10⁸ ÷ (4/3) = 2.25 × 10⁸ m/s. A related one asks for the refractive index of glass with respect to water: n_gw = n_g/n_w = (3/2)/(4/3) = 9/8 = 1.125. The rule is that the refractive index of A with respect to B equals n_A/n_B, absolute values.

A lens is a transparent material bound by two surfaces, of which at least one is spherical. A convex lens is thicker in the middle than at the edges; it bulges outwards on both sides (double convex) and converges parallel rays, so it is a converging lens. A concave lens is thinner in the middle than at the edges, curves inwards on both sides (double concave) and diverges parallel rays, so it is a diverging lens. A lens has two spherical surfaces, hence two centres of curvature, C₁ and C₂, and two radii of curvature.

Terms, in NCERT wording. The principal axis is an imaginary straight line passing through the two centres of curvature. The optical centre (O) is the central point of the lens; a ray passing through the optical centre goes without any deviation. The aperture is the effective diameter of the circular outline of the lens; we deal only with thin lenses of small aperture. The principal focus of a convex lens is the point on the principal axis where rays parallel to the axis converge after refraction; for a concave lens it is the point on the principal axis from which the parallel rays appear to diverge after refraction. Because light can fall on either face, a lens has two principal foci, F₁ and F₂, at equal distances on either side. The focal length f is the distance of the principal focus from the optical centre.

The NCERT activity for measuring the focal length of a convex lens: hold the lens facing the sun, catch the light on a paper, and adjust until you get the sharpest bright spot; the distance of the paper from the lens is approximately the focal length. Do not look at the sun through the lens.

The MCQ which material cannot be used to make a lens has the answer clay, since a lens must be transparent. Another frequent one asks which lens you would use to read the small letters in a dictionary: a convex lens of short focal length (5 cm), because a short focal length gives greater magnification when the object is held inside F.

The three rays used for lens diagrams: (1) a ray parallel to the principal axis passes through F₂ on the other side after refraction through a convex lens, or appears to diverge from F₁ on the same side for a concave lens; (2) a ray passing through F₁ of a convex lens (or appearing to meet at F₂ for a concave lens) emerges parallel to the principal axis; (3) a ray through the optical centre passes straight through, undeviated. As with mirrors, any two are enough.

Convex lens, six positions, matching the concave mirror table with 2F in place of C:

• Object at infinity: image at F₂, highly diminished, real, inverted.
• Object beyond 2F₁: image between F₂ and 2F₂, diminished, real, inverted.
• Object at 2F₁: image at 2F₂, same size, real, inverted.
• Object between F₁ and 2F₁: image beyond 2F₂, enlarged, real, inverted.
• Object at F₁: image at infinity, highly enlarged, real, inverted.
• Object between O and F₁: image on the same side as the object, enlarged, virtual, erect. This is the magnifying glass.

Concave lens, two positions: object at infinity gives a point image at F₁, virtual and erect, highly diminished; object anywhere between infinity and O gives a virtual, erect, diminished image between F₁ and O on the same side as the object. A concave lens always forms a virtual, erect, diminished image, whatever the object distance, which parallels the convex mirror.

Drawing for full marks: axis, O on the lens, F₁, 2F₁ on the left and F₂, 2F₂ on the right at equal spacing, object arrow on the left, two rays from the tip with arrowheads, image where the refracted rays meet (solid arrow for real, dotted for virtual with dotted backward extensions). For the concave lens, the refracted rays diverge; extend the first one backwards through F₁ and let it meet the undeviated central ray.

The half-covered lens (NCERT exercise 9, CBSE 2024 MCQ): if one half of a convex lens is covered with black paper, the lens still forms a complete image of the object, because every part of the lens refracts rays from every point of the object; only the intensity (brightness) of the image is reduced, since fewer rays now reach it.

Lenses use the same New Cartesian Sign Convention as mirrors, with one change of reference point: all distances are measured from the optical centre of the lens instead of the pole. The object is on the left, so u is negative. Distances to the right of O, along the direction of the incident light, are positive; distances to the left are negative. Heights above the axis are positive, below the axis negative.

The consequences you must carry into every numerical. A convex lens has its principal focus F₂ on the right, so its focal length is positive. A concave lens has the focus from which rays appear to diverge on the left (F₁), so its focal length is negative. A real image formed by a convex lens is on the right, so v is positive; a virtual image (magnifying glass, or any concave-lens image) is on the left, on the same side as the object, so v is negative. An inverted image has negative height.

Compare with mirrors carefully, because the pairing confuses students under exam pressure. For a mirror, a real image is in front (v negative) and a virtual image is behind (v positive). For a lens it is the other way round: a real image is on the far side (v positive) and a virtual image is on the same side as the object (v negative). The reason is simply that light passes through a lens but bounces off a mirror, so the region the light ends up in is on opposite sides in the two cases.

The MCQ a spherical mirror and a thin spherical lens have each a focal length of −15 cm; the mirror and the lens are likely to be, is answered both concave. A lens of positive focal length is convex; a lens of negative power is concave. Negative signs on u are automatic; the sign of f is decided by the type of lens; the sign of v comes out of the formula and tells you which side and hence whether the image is real or virtual.

The lens formula is 1/v − 1/u = 1/f. Note the minus sign; it is the one difference from the mirror formula, and mixing the two is the most common error in the numericals. It holds for convex and concave lenses in all situations, with the sign convention applied.

Magnification for a lens is m = h′/h = v/u, without the minus sign that the mirror version has. The interpretation is the same as for mirrors: negative m means real and inverted, positive m means virtual and erect; |m| greater than 1 means enlarged. So for a convex lens with the object beyond F, u is negative and v is positive, giving m negative, which is the real inverted image; for the magnifying glass both u and v are negative, m is positive and greater than 1, the virtual, erect, enlarged image.

Worked example (NCERT exercise 10, asked in the same form in boards). An object 5 cm long is held 25 cm from a converging lens of focal length 10 cm. u = −25 cm, f = +10 cm, h = +5 cm. 1/v = 1/f + 1/u = 1/10 − 1/25 = (5 − 2)/50 = 3/50, so v = 50/3 = +16.7 cm. The image is 16.7 cm on the other side of the lens, real and inverted. m = v/u = 16.7/(−25) = −0.67, so h′ = −0.67 × 5 = −3.3 cm: inverted, 3.3 cm tall, diminished. The object is beyond 2F (20 cm), so the table predicts an image between F and 2F, diminished, real, inverted: consistent.

Second worked example, concave lens (NCERT exercise 11). A concave lens of focal length 15 cm forms an image 10 cm from the lens. The image of a concave lens is virtual and on the object side, so v = −10 cm, f = −15 cm. 1/u = 1/v − 1/f = −1/10 + 1/15 = (−3 + 2)/30 = −1/30, so u = −30 cm: the object is 30 cm in front of the lens. m = v/u = (−10)/(−30) = +1/3, virtual, erect, one-third size.

Third, the magnifying-glass type (2020 palmist question). A convex lens of focal length 10 cm is used to see a virtual image magnified two times. m = +2 = v/u, so v = 2u. Then 1/(2u) − 1/u = 1/10, giving −1/(2u) = 1/10, so u = −5 cm and v = −10 cm. The object is held 5 cm from the lens, inside F, and the image is 10 cm away on the same side.

The ability of a lens to converge or diverge light rays is expressed by its power. A convex lens of short focal length bends rays through large angles and converges them close to the lens; a concave lens of short focal length diverges them strongly. So power is defined as the reciprocal of the focal length: P = 1/f. With f in metres, the SI unit of power is the dioptre, symbol D. One dioptre is the power of a lens whose focal length is 1 metre: 1 D = 1 m⁻¹. This definition of the dioptre is a standard 1- or 2-mark question.

Signs follow from the focal length. A convex lens has positive focal length and positive power; a concave lens has negative focal length and negative power. Opticians prescribe lenses by power: a prescription of +2.0 D means a convex lens of focal length 1/2 = 0.5 m = 50 cm, and −2.5 D means a concave lens of focal length 1/(−2.5) = −0.4 m, that is, 40 cm.

Worked examples. Find the focal length of a lens of power −2.0 D: f = 1/P = 1/(−2.0) = −0.5 m = −50 cm, a concave lens. A doctor prescribes +1.5 D: f = 1/1.5 = 0.667 m = 66.7 cm, a converging (convex) lens. A concave lens of focal length 2 m has P = 1/(−2) = −0.5 D. A convex lens forms a real, inverted image of a needle at 50 cm from the lens, same size as the needle: the same-size real image means the object is at 2F and the image at 2F, so 2f = 50 cm, f = 25 cm = 0.25 m, the needle is 50 cm in front, and P = 1/0.25 = +4 D.

Combination of lenses. When several thin lenses are placed in contact, the power of the combination is the algebraic sum of the individual powers: P = P₁ + P₂ + P₃ + … Opticians use this to test a patient's eye with a stack of lenses, and it is why a convex lens of +10 D placed in contact with a concave lens of −10 D behaves like a plane glass slab of zero power. Numericals give two focal lengths in centimetres; convert to metres, take the reciprocals, add, and then take the reciprocal again for the combined focal length. Two lenses of focal lengths +20 cm and −50 cm give P = 5 − 2 = +3 D, so the combination acts as a converging lens of focal length 33.3 cm.